Instabilities of charged black holes and small hairy black holes

Reissner-Nordstrom black holes (RN BH) in AdS suffer from two linear instabilities, superradiance and the near horizon instability. The endpoint of these instabilities is a hairy black hole. I will present an analytic perturbative construction of these hairy black holes and analyse the phase space of solutions. In asymptotically flat spacetime, inspired by the black hole bomb of Press and Teukolsky, one can surround a RN BH by a reflecting box. It is known that this system suffers from superradiant instability. We show that the near horizon instability is also present there. As in the AdS case, the endpoint is a hairy black hole that we construct analytically in a perturbative expansion. We comment on the matter content of the box.